Some new sufficient conditions for $2p$-Hamilton-biconnectedness of graphs

A balanced bipartite graph $G$ is said to be $2p$-Hamilton-biconnected if for any balanced subset $W$ of size $2p$ of $V(G)$, the subgraph induced by $V(G)\mathrm{\backslash }W$ is Hamilton-biconnected. In this paper, we prove that “Let $p\ge 0$ and $G$ be a balanced bipartite graph of order $2n$ with minimum degree $\delta (G)\ge k$, where $n\ge 2k-p+2$ and $k\ge p$. If the number of edges $e(G)>n(n-k+p-1)+(k+2)(k-p+1),$ then $G$ is $2p$-Hamilton-biconnected except some exceptions.” Furthermore, this result is used to present two new spectral conditions for a graph to $2p$-Hamilton-biconnected. Moreover, the similar results are also presented for nearly balanced bipartite graphs.